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The binomial series gives $F(A,C;C;u)=(1-u{)}^{-A}$ wherever the defining series converges, followed by analytic continuation.

Away from resonant parameter cases, two independent local solutions at infinity areand the expression obtained by exchanging $a$ and $b$. Every analytic continuation of a hypergeometric solution into their common domain is a constant linear combination of this basis.

When $C$ is not an integer, a second local solution at zero is $u^{1-C}F(A-C+1,B-C+1;2-C;u)$.

With compatible branches and initially near $z=0$,It follows by applying a Möbius transformation and a dependent-variable rescaling to the same Papperitz symbol, then matching the normalized analytic solution at zero.

BRST symmetry is a nilpotent fermionic symmetry of a gauge-fixed action. Its differential $s$ replaces an infinitesimal gauge parameter by the ghost field and satisfies $s^2=0$.

An axial gauge imposes $n^{\mu }{A}_{\mu }=0$ for a fixed vector $n^{\mu }$. Its Faddeev-Popov operator is $n^{\mu }{D}_{\mu }$; on the strict gauge slice its gauge-field-dependent part vanishes, so its ghosts decouple.

The Faddeev-Popov determinant is the functional Jacobianthat compensates for the change from integration along a gauge orbit to a gauge-fixing condition $G(A)=0$. It can be represented by a path integral over a Faddeev-Popov ghost field pair.

Adding $-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\alpha )$ gives a family of Lorentz-covariant gauges; $\alpha =1$ is Feynman gauge.

The beta--gamma identity gives $B(z,z)=\Gamma (z{)}^2/\Gamma (2z)\sim 2/z$ as $z\to 0$ through the right half-plane.

For a symmetric random walk started at $i\in \{-a,\dots ,b\}$ and stopped on first reaching $-a$ or $b$, the mean duration isIn particular, ${\mathbb{E}}_{0}T=ab$. This follows either from the first-step recurrence or by stopping the martingale $S_n^2-n$.

The Galactic scale height measures the vertical thickness of a tracer. Gravity, velocity dispersion, thermal pressure, magnetic and cosmic-ray support, particle transport, and source lifetime determine its value.

Diffuse Galactic gamma rays arise mainly from neutral-pion decay after cosmic-ray nuclear collisions, electron bremsstrahlung in gas, and inverse-Compton scattering from radiation fields.

Galactic synchrotron emission is nonthermal radio radiation from relativistic electrons spiralling in the Galactic magnetic field. Its broad latitude distribution reflects cosmic-ray transport and the magnetic halo.

If $g\in L^1(a,b)$, then $F(x)=F(a)+{\int }_a^{x}g(s)ds$ is absolutely continuous and satisfies $F^{\prime }=g$ almost everywhere. Conversely, every absolutely continuous function has this form.

If two unital operations on one set share their identity and satisfy $(a\ast b)\cdot (c\ast d)=(a\cdot c)\ast (b\cdot d)$, then the operations coincide and are commutative.

A functor $F:\mathcal{C}\to \mathcal{D}$ is full and faithful when every mapis a bijection.

The functor category has functors $\mathcal{C}\to \mathcal{D}$ as objects and natural transformations as morphisms.

If $K/k$ is a finitely generated field extension, write $K=\mathrm{Frac}(A)$ for a finitely generated integral $k$-algebra $A$. The projective closure of the affine variety $\mathrm{Spec}A$ is an irreducible projective variety with function field $K$.

For an irreducible variety $X$ over a field $k$,Consequently birational irreducible varieties have the same dimension.

If $R$ is a nonzero commutative ring and $R^m\cong R^n$, reduction modulo a maximal ideal produces isomorphic vector spaces of dimensions $m$ and $n$. Hence $m=n$.